A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current I. The distance between the wire and the left side of the loop is r. The loop is pulled to the right as indicated.

What is the magnitude of the net force on the loop when the induced current is i?

Electromagnetism}Magnetic Force

The magnetic force of a wire is given by , where I is the current of the wire and l its length.

The field that produces the force on the loop is given by the long wire (see the previous problem for why). The field of that wire is given trivially by Ampere's Law to be , where r is the radial distance away from its center.

Only two wires from the loop contribute to the force, since the cross-product yields 0 force for the two horizontal components. Thus, the net force on the loop with current i with vertical components of length b is . Combine the fraction to get choice (D).

Alternate Solutions

h.fei102012-11-04 05:11:22

The force should be proportional to the length b, remember the formula F=BIL in high school, so eliminate A,B,E.

Here is one more way of looking at this problem.
In the limit a->0, the net force on the wire must be zero, because you will have two equal and opposite forces acting on the wire. So, the only choice that goes to zero in the limit a->0 is D. So D is the correct choice:)

neon372010-11-12 00:56:42

incorrect. Also, A and B go to zero as ln(1)=0. This works only if you eliminate A and B first.

Here is one more way of looking at this problem.
In the limit a->0, the net force on the wire must be zero, because you will have two equal and opposite forces acting on the wire. So, the only choice that goes to zero in the limit a->0 is D. So D is the correct choice:)

If the area of the loop is zero, there will be no flux, and hence no induced current and no force. So, F=0 if a=0. This eliminates A, B, and C. Also, F=0 if b=0. This eliminates E. Only choice D is left.

spacebabe472006-11-01 15:23:30

Edit:

If the area of the loop is zero, there will be no flux, and hence no induced current and no force. So, F=0 if a=0. This eliminates C. Also, F=0 if b=0. This eliminates A, B, and E. Only choice D is left.

You can also consider the limiting case where the force will go to zero as r goes to zero. This will eliminate choices C and E. From here you can make an educated guess. One would expect the force to depend on both dimensions a and b. So then chose choice D.